Energy Conservation Method

Q. A solid cylinder (radius $r=0.2\text{m}$) rolls without slipping in a cylindrical trough (radius $R=0.8\text{m}$). The time period of small oscillations is
[Take $g=10{\text{m/s}}^2$]

1. $\frac{3\pi }{5}\text{sec}$
2. $\frac{\pi }{5}\text{sec}$
3. $\frac{2\pi }{5}\text{sec}$
4. $\frac{\pi }{3}\text{sec}$

Q. Two blocks A and B, each of mass m, are connected by a massless spring of natural length L and spring constant k. the blocks are initially resting on a smooth horizontal floor with the spring at its natural length. A third identical block C, also of mass m, moving on the floor with a speed along the line joining A and B, collides with A (see figure) Then

1. The kinetic energy of A – B system at maximum compression of the spring is zero
2. The kinetic energy of A – B system at maximum compression of the spring is $\frac{m{v}^2}{4}$
3. The maximum compression of the spring is $v\sqrt{\frac{m}{k}}$
4. The maximum compression of the spring is $v\sqrt{\frac{2m}{k}}$
   Let x be the maximum compression of the spring. From the law of conservation of energy, we have $\frac{1}{2}m{v}^2=m{u}^2+\frac{1}{2}k{x}^2$

Q. One end of a long metallic wire of length L is tied to the ceiling. The other end is tied to a massless spring of spring constant K. A mass m hangs freely from the free end of the spring. The area of cross – section and Young’s modulus of the wire are A and Y respectively. If the mass is slightly pulled down and released, it will oscillate with a time period given by

1. $2\pi \sqrt{\frac{m}{k}}$
2. $2\pi {\left[\frac{(YA+KL)m}{YAK}\right]}^{\frac{1}{2}}$
3. $2\pi \left(\frac{mYA}{KL}\right)$
4. $2\pi \left(\frac{mL}{YA}\right)$